Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#13127 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C7 | #388 |
| commutator | C22×C4 | C22×C4×C7 |
| Frattini, cosocle | C22×C8 | C23×C7 |
| socle | C2×C7 | #1924 |
| prime radical | C2×C7×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C7 ◁ C22×C7×C8 ◁ G | |
300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7)(8,9,13,19,12,17,21,16,14,18,30,36,28,37,11,26)(10,20,22,35,24,15,27,33,25,34,29,38,23,32,39,31);(8,10,21,39,28,23,13,29,14,25,11,27,12,24,30,22)(9,15,16,35,37,20,19,31,18,32,26,38,17,34,36,33);(8,11,12,13,14,21,28,30)(9,16,17,36,18,26,37,19)(10,22,23,39,25,29,24,27)(15,31,20,38,32,33,34,35)322 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 6115);